On the relation between tonal and broadband content of hull pressure spectra due to cavitating ship propellers

Transcription

1 Joul of Physics: Cofeece Seies PAPER OPEN ACCESS O the eltio betwee tol d bodbd cotet of hull pessue spect due to cvittig ship popelles To cite this ticle: J Bossches 015 J. Phys.: Cof. Se View the ticle olie fo updtes d ehcemets. This cotet ws dowloded fom IP ddess o 05/11/018 t 14:33

2 9th Itetiol Symposium o Cvittio (CAV015) Joul of Physics: Cofeece Seies 656 (015) IOP Publishig doi: / /656/1/01101 O the eltio betwee tol d bodbd cotet of hull pessue spect due to cvittig ship popelles J Bossches MARIN, Hgsteeg, 6700 AA, Wgeige, the Nethelds E-mil: j.bossches@mi.l Abstct. Cvittig popelles geete pessue fluctutios o the hull of the ship. These pessue fluctutios e usully lyzed i the fequecy domi usig FFTs d the spectum is composed of tols t multiples of the blde pssge fequecy d bodbd pt. The two e ofte cosideed septely but eltio betwee the two exists which hs bee ivestigted by theoeticl sigl lysis. It will be show tht the bodbd pt is elted to the vibility of the sigl betwee blde pssges d simple pocedue is poposed to qutify the vibility i tems of mplitude d phse gle. The pocedue hs bee pplied to dt set obtied t se tils. 1. Itoductio Cvittig popelles my geete ibod oise d vibtio hidce fo cew d psseges obod ships. Ofte, it is the hull pessue spectum tht is lyzed to evlute the hull-excittio foce of the cvittig popelle. This spectum is composed of tol compoets t hmoics of the blde pssge fequecy d bodbd pt. The bodbd pt is kow to be cused by the vibility of the pessue sigl [1][], but qutittive desciptio is missig i litetue. The peset ppe pesets fomultio tht c fill this gp lthough mthemticl poof is still to be completed. Befoe pesetig the fomultio some bsic spects of the spectum e discussed.. Stylised spect of sigle blde pssge Stylised hull pessue time tces d coespodig spect of cvity collpse duig sigle blde pssge e show i figue 1. The sigl ws geeted usig the bsolute vlue of sie fo umbe of cycles. Sigl 1 d 1b hve o dmpig d the width of the bodbd hump i the spectum, with cete fequecy t 10 Hz which is the epet fequecy of the pessue pulse, is ivesely popotiol to the umbe of cycles. Fo sigl d b expoetil dmpig ws dded to the time tce which leds to smooth vitio of the spectum fom fequecy zeo to the bodbd hump. 3. Stylised spect of multiple blde pssges Alyticl fomultios fo the effect of mplitude d phse gle modultio o spect of pulse tis, efeed to s blde pssges i the peset ppe, hve bee peseted by McFle [3] d the theoy is biefly estted hee. Coside tht the mplitude desity spectum of sigle blde pssge t time zeo is give by. Cotet fom this wok my be used ude the tems of the Cetive Commos Attibutio 3.0 licece. Ay futhe distibutio of this wok must miti ttibutio to the utho(s) d the title of the wok, joul cittio d DOI. Published ude licece by IOP Publishig Ltd 1

3 9th Itetiol Symposium o Cvittio (CAV015) Joul of Physics: Cofeece Seies 656 (015) IOP Publishig doi: / /656/1/01101 ) b) Figue 1. Stylised pessue time tces (left) d omlised powe spectl desities (ight) due to cvity collpse d ebouds duig sigle blde pssge The spectum, S, of sum of N 1 (o-ovelppig) blde pssges spced T pt, with mplitude d eltive time shift t, is the give by N exp S i T t N d the powe spectl desity (psd) is give by 1 (1) R S N T. With mplitude modultio the vlues fo e sttisticlly descibed by the me d stdd devitio while thee is o eltive time shift t 0. Afte some lgebic mipultios i which ifiite seies of blde pssges is ssumed, we fid the psd fo mplitude modultio, whee T R T coespods to the blde pssge fequecy (bpf) d () to Dic comb with fequecy spcig. The spectum is composed of bodbd pt with mplitude with tol pt with mplitude weighted with the fequecy bi width. Phse modultio is defied hee s bity vitio i ivl time T sigl fo ech blde pssge while the mplitude does ot vy i time summed t of the pessue. The pobbility of t is give by ussi distibutio q with zeo me d stdd devitio p, 1 qt exp t p. (3) p Afte legthy lgebic mipultios, the followig expessio fo the psd fo phse gle modultio is obtied, R 1 exp p exp p (4) T The effect of the mplitude d phse gle modultio is illustted i figue, followig []. A stylised spectum is sketched of sigle blde pssge togethe with the spectum of seies of blde pssges. The fequecy is o-dimesiolized by the bpf. If the sigl epet pefectly thee is o

4 9th Itetiol Symposium o Cvittio (CAV015) Joul of Physics: Cofeece Seies 656 (015) IOP Publishig doi: / /656/1/01101 modultio d the spectum cosists of tols t hmoics of the bpf. Rdom vitios i mplitude loe betwee diffeet blde pssges hdly ffect the mplitudes of the tols. Howeve, the spectum ow does coti bodbd pt of which the mgitude is popotiol to the stdd devitio of the mplitude vitios. Rdom chges i time of ivl of the sigl led to dom phse gle vitios of which the mgitude iceses liely with fequecy. This dom phse modultio deceses the mplitude of the tols d edistibutes the powe ove bodbd egio. The spedig is most poouced t highe fequecies d my cuse the complete disppece of the tol. Figue. Stylised pessue time tces d esultig spect fo multiple blde pssges with mplitude d phse modultio 4. Pcticl exmple of combied mplitude d phse gle modultio Fo the combitio of mplitude d phse gle modultio, the equtios () d (4) e combied i heuistic me with mthemticl poof still to be completed. The fomultio is give by 1 exp exp R p p (5) T I elity, the shpe of the spectum lso vies with the blde pssges d the stdd devitio fo mplitude d phse is fequecy depedet. The fomultio hs bee pplied to the hull pessue sigl geeted by fou blded cvittig popelle duig se til. Dt sets e lyzed by subdividig the time tce ito shft evolutios with shft evolutio defied by costt umbe of smples which does ot chge i time, see figue 3. This implies tht vitios of the shft evolutio te i time due to se stte e itepeted s phse gle modultio. Ech shft evolutio is lyzed with FFT, fte which the me d stdd devitio of both mplitude d phse gle c be computed fo ech hmoic of the shft te fequecy. The vitio of stdd devitio of mplitude d phse gle with fequecy is give i figue 3c d 3d. The phse gle ws uwpped which explis vlues fo the stdd devitio of the phse lge th 360 deg. 3

5 9th Itetiol Symposium o Cvittio (CAV015) Joul of Physics: Cofeece Seies 656 (015) IOP Publishig doi: / /656/1/01101 The spectum tht is geeted fom the mplitude ifomtio loe by usig equtio () is efeed to s AM d the esult is give i figue 3b. To simplify the gph, the tols e give fo the hmoics of the bpf oly. Usig equtio (5), we c ecostuct the spectum with mplitude d phse modultio (desigted APM i figue 3b) d compe this with the efeece spectum tht is obtied by tkig FFT ove lge umbe of evolutios. Fo simplicity, lie fit of the stdd devitio of the phse gles t hmoics of the bpf is tke except fo the fist bpf. The pessue mplitude t the fist bpf is lso due to the pssge of the o-cvittig blde which epets well s show by the smll vlues of the eltive stdd devitio. Figue 3b shows esoble geemet betwee the two spect with some diffeeces t the d d 3 d bpf suggestig tht, fte some futhe impovemet, the poposed fomultio d lysis pocedue my be useful fo qutifyig the eltio betwee highe ode tols d bodbd cotet of hull pessue spect. ) Time tce of pessue sigl b) Spect c) Vitio of stdd devitio of the d) Vitio of the stdd devitio of the mplitude with fequecy phse gle with fequecy Figue 3. Alysis of the hull pessue sigl of (fou blded) cvittig popelle s mesued o se til Refeeces [1] Bk 1988 O the mechisms of popelle cvittio oise (Ph.D. thesis, Chlmes Uivesity of Techology, ötebog, Swede) [] Bossches J 009 Ivestigtio of hull pessue fluctutios geeted by cvittig votices, Fist Itetiol Symposium o Mie Populsos, smp 09, Todheim, Nowy [3] McFle 1949 O the eegy-spectum of lmost peiodic successio of pulses, Poc. of the Ist. of Rdio Eg., Vol. 37, No. 10, pp